Inequalities for generalized Riemann–Liouville fractional integrals of generalized strongly convex functions

Some new integral inequalities for strongly $(\alpha ,h-m)$ ( α , h − m ) -convex functions via generalized Riemann–Liouville fractional integrals are established. The outcomes of this paper provide refinements of some fractional integral inequalities for strongly convex, strongly m-convex, strongly $(\alpha ,m)$ ( α , m ) -convex, and strongly $(h-m)$ ( h − m ) -convex functions. Also, the refinements of error estimations of these inequalities are obtained by using two fractional integral identities. Moreover, using a parameter substitution and a constant multiplier, k-fractional versions of established inequalities are also given.

An approach in studying gyrostat motion with variable gyrostatic moment

The problem of the motion of a gyrostat with a fixed point and a variable gyrostatic moment under the action of gravity force is considered. A new method for integrating the equations of motion of a system consisting of a carrier body and three rotors that rotate around the main axes is proposed. The method can be attributed to the method of variation of the constant in the function for the gyrostatic moment, which linearly depends on the vector of vertical. In case of a constant multiplier, the gyrostatic moment satisfies the Poisson equation, and its variation is found from the integral of areas. The original equations have been reduced to a fifth-order system. New solutions of these equations are obtained in the case of a spherical mass distribution for the gyrostat and for the precessional motions of a carrier body. An explicit form of the gyrostatic moment is established for the case of three invariant relations.

Distribution of the Proportionality Coefficient of Winter Route Accounting

In previous studies it was shown that the coefficient of proportionality of the winter route count (WRC) of animals included in the formula of WRC in the form of a constant multiplier π/2, is actually a random variable – the same as the average number of intersections account route traces per unit length, and the average length of the diurnal animals. The value π/2 is the mathematical expectation value of the proportionality factor, provided that the count route equiprobably crosses the daily footprint at any place and at any angle from 0 to 2π during a winter route counting of animals. At the same time, both the nature of the distribution of the coefficient as a random variable and the values of its variance as its other statistical characteristics remained unknown. In this study, it was found that when the above-mentioned count conditions are met, the distribution of the proportionality coefficient of WRC as a random variable will be exponential or power-like. This allows calculating the values of its variance and relative statistical error in advance without collecting additional count data.

Pengembangan Bagan Kendali 3 Sigma Berbasis Simulasi Untuk Pengawasan Variabilitas Proses Multivariat

This article describes a method in developing control charts for generalized variance as a quality statistics in terms of process variability through simulation. Mathematical equation that maps sample size n and number of quality variables p onto constant multiplier of standard deviation K is obtained thorough least square method using simulated data. The constant K for a certain n and p is used for control charts with the upper control limits of UCL=μ+Kσ where μ and σ are the mean and the standard deviation of the generalized variance, respectively. The simulation of finding the constant K is used with the constraint of 3 sigma paradigm.

CONIC CPPIs

Constant proportion portfolio insurance (CPPI) is a structured product created on the basis of a trading strategy. The idea of the strategy is to have an exposure to the upside potential of a risky asset while providing a capital guarantee against downside risk with the additional feature that in case the product has since initiation performed well more risk is taken while if the product has suffered mark-to-market losses, the risk is reduced. In a standard CPPI contract, a fraction of the initial capital is guaranteed at maturity. This payment is assured by investing part of the fund in a riskless manner. The other part of the fund’s value is invested in a risky asset to offer the upside potential. We refer to the floor as the discounted guaranteed amount at maturity. The percentage allocated to the risky asset is typically defined as a constant multiplier of the fund value above the floor. The remaining part of the fund is invested in a riskless manner. In this paper, we combine conic trading in the above described CPPIs. Conic trading strategies explore particular sophisticated trading strategies founded by the conic finance theory i.e. they are valued using nonlinear conditional expectations with respect to nonadditive probabilities. The main idea of this paper is that the multiplier is taken now to be state dependent. In case the algorithm sees value in the underlying asset the multiplier is increased, whereas if the assets is situated in a state with low value or opportunities, the multiplier is reduced. In addition, the direction of the trade, i.e. going long or short the underlying asset, is also decided on the basis of the policy function derived by employing the conic finance algorithm. Since nonadditive probabilities attain conservatism by exaggerating upwards tail loss events and exaggerating downwards tail gain events, the new Conic CPPI strategies can be seen on the one hand to be more conservative and on the other hand better in exploiting trading opportunities.